# Terminology

The following terms, symbols, and decorators are used in text and diagrams throughout this guide.

## Notation

- Bold face variables indicate vectors or matrices and non-bold face variables represent scalars. 
- The default frame for each variable is the local frame: $\ell{}$.
  Right [superscripts](#superscripts) represent the coordinate frame.
  If no right superscript is present, then the default frame $\ell{}$ is assumed.
  An exception is given by Rotation Matrices, where the lower right subscripts indicates the current frame and the right superscripts the target frame.
- Variables and subscripts can share the same letter, but they always have different meaning.

## Acronyms

Acronym | Expansion
--- | ---
AOA | Angle Of Attack. Also named *alpha*.
AOS | Angle Of Sideslip. Also named *beta*.
FRD | Coordinate system where the X-axis is pointing towards the Front of the vehicle, the Y-axis is pointing Right and the Z-axis is pointing Down, completing the right-hand rule.
FW | Fixed-Wing.
MC | MultiCopter.
MPC or MCPC | MultiCopter Position Controller. MPC is also used for Model Predictive Control.
NED | Coordinate system where the X-axis is pointing towards the true North, the Y-axis is pointing East and the Z-axis is pointing Down, completing the right-hand rule.
PID | Controller with Proportional, Integral and Derivative actions.


## Symbols

Variable | Description
--- | ---
$x,y,z$ | Translation along coordinate axis x,y and z respectively.
$\boldsymbol{\mathrm{r}}$ | Position vector: $\boldsymbol{\mathrm{r}} = [x \quad y \quad z]^{T}$
$$\boldsymbol{\mathrm{v}}$$ | Velocity vector: $\boldsymbol{\mathrm{v}} = \boldsymbol{\mathrm{\dot{r}}}$
$$\boldsymbol{\mathrm{a}}$$ | Acceleration vector: $\boldsymbol{\mathrm{a}} = \boldsymbol{\mathrm{\dot{v}}} = \boldsymbol{\mathrm{\ddot{r}}}$
$$\alpha$$ | Angle of attack (AOA).
$$b$$ | Wing span (from tip to tip).
$$S$$ | Wing area.
$$AR$$ | Aspect ratio: $AR = b^2/S$
$$\beta$$ | Angle of sideslip (AOS).
$$c$$ | Wing chord length.
$$\delta$$ | Aerodynamic control surface angular deflection. A positive deflection generates a negative moment.
$$\phi,\theta,\psi$$ | Euler angles roll (=Bank), pitch and yaw (=Heading).
$$\Psi$$ | Attitude vector: $\Psi = [\phi \quad \theta \quad \psi]^T$
$$X,Y,Z$$ | Forces along coordinate axis x,y and z.
$$\boldsymbol{\mathrm{F}}$$| Force vector: $\boldsymbol{\mathrm{F}}= [X \quad Y \quad Z]^T$
$$D$$ | Drag force.
$$C$$ | Cross-wind force.
$$L$$ | Lift force.
$$g$$ | Gravity.
$$l,m,n$$ | Moments around coordinate axis x,y and z.
$$\boldsymbol{\mathrm{M}}$$ | Moment vector $\boldsymbol{\mathrm{M}} = [l \quad m \quad n]^T$
$$M$$ | Mach number. Can be neglected for scale aircraft.
$$\boldsymbol{\mathrm{q}}$$ | Vector part of Quaternion.
$$\boldsymbol{\mathrm{\tilde{q}}}$$ | Hamiltonian attitude quaternion. $\boldsymbol{\mathrm{\tilde{q}}} = (q_0, q_1, q_2, q_3) = (q_0, \boldsymbol{\mathrm{q}})$.<br> $\boldsymbol{\mathrm{\tilde{q}}}{}$ describes the attitude relative to the local frame $\ell{}$. To represent a vector in local frame given a vector in body frame, the following operation can be used:  $\boldsymbol{\mathrm{\tilde{v}}}^\ell = \boldsymbol{\mathrm{\tilde{q}}} \, \boldsymbol{\mathrm{\tilde{v}}}^b \, \boldsymbol{\mathrm{\tilde{q}}}^*{}$ (or $\boldsymbol{\mathrm{\tilde{q}}}^{-1}{}$ instead of $\boldsymbol{\mathrm{\tilde{q}}}^*{}$ if $\boldsymbol{\mathrm{\tilde{q}}}{}$ is not unitary). $\boldsymbol{\mathrm{\tilde{v}}}{}$ represents a *quaternionized* vector: $\boldsymbol{\mathrm{\tilde{v}}} = (0,\boldsymbol{\mathrm{v}})$
$$\boldsymbol{\mathrm{R}}_\ell^b$$ | Rotation matrix. Rotates a vector from frame $\ell{}$ to frame $b{}$. $$\boldsymbol{\mathrm{v}}^b = \boldsymbol{\mathrm{R}}_\ell^b \boldsymbol{\mathrm{v}}^\ell$$
$$\Lambda$$ | Leading-edge sweep angle.
$$\lambda$$ | Taper ratio: $\lambda = c_{tip}/c_{root}$
$$w$$ | Wind velocity.
$$p,q,r$$ | Angular rates around body axis x,y and z.
$$\boldsymbol{\omega}^b$$ | Angular rate vector in body frame: $\boldsymbol{\omega}^b = [p \quad q \quad r]^T$
$$\boldsymbol{\mathrm{x}}$$ | General state vector.

### Subscripts / Indices

Subscripts / Indices | Description
--- | ---
$$a$$ | Aileron.
$$e$$ | Elevator.
$$r$$ | Rudder.
$$Aero$$ | Aerodynamic.
$$T$$ | Thrust force.
$$w$$ | Relative airspeed.
$$x,y,z$$ | Component of vector along coordinate axis x, y and z.
$$N,E,D$$ | Component of vector along global north, east and down direction.

<a id="superscripts"></a>
### Superscripts / Indices

Superscripts / Indices | Description
--- | ---
$$\ell$$ | Local-frame. Default for PX4 related variables.
$$b$$ | Body-frame.
$$w$$ | Wind-frame.


## Decorators

Decorator | Description
--- | ---
$$()^*$$ | Complex conjugate.
$$\dot{()}$$ | Time derivative.
$$\hat{()}$$ | Estimate.
$$\bar{()}$$ | Mean.
$$()^{-1}$$ | Matrix inverse.
$$()^T$$ | Matrix transpose.
$$\tilde{()}$$ | Quaternion.

